To solve the problem, we need to determine the perimeter of the shaded portion in the given figure. The shaded region consists of a semicircular arc and two straight line segments.
1. Understanding the Geometry:
The figure shows a semicircle with diameter . The points and are endpoints of the diameter, and is a point on the semicircle such that and . The shaded region includes the semicircular arc from to and the two straight line segments and .
2. Calculating the Diameter :
Since and are segments of the semicircle, the total length of the diameter is:
Thus, the radius of the semicircle is:
3. Calculating the Length of the Semicircular Arc:
The circumference of a full circle is given by . For a semicircle, the arc length is half of the circumference:
Given that , the arc length is:
4. Calculating the Perimeter of the Shaded Region:
The perimeter of the shaded region consists of the semicircular arc and the two straight line segments and . Therefore, the total perimeter is:
Substituting the known values:
Final Answer:
The perimeter of the shaded portion is .
